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. 436.
[Philosophical Magazine, Vol. xxxvi. pp. 429449, 1918.]
RECENT researches have emphasized the importance of a clear comprehension of the operation under various conditions of a group of similar unit sources, or centres, of iso-periodic vibrations, e.g. of sound or of light. The sources, supposed to be concentrated in points, may be independently excited (as probably in a soda flame), or they may be constituted of similar small obstacles in an otherwise uniform medium, dispersing plane waves incident upon them. We inquire into an effect, such as the intensity, at a great distance from the cloud, either in a particular direction, or in the average of all directions. For convenience of calculation and statement we shall consider especially sonorous vibrations; but most of the results are equally applicable to electric vibrations, as in light, the additional complication being merely such as arises from the vibrations being transverse to the direction of propagation.
If the centres, supposed to be distributed at random in a region whose three dimensions are all large, are spaced widely enough in relation to the wave-length (A,) to act independently, the question reduces itself to one formerly treated*, for it then becomes merely one of the composition of a large number (n) of unit vibrations of arbitrary phases. It is known that the "expectation" of intensity in any direction is n times that due to a single centre, or (as we may say) is equal to n. The word " expectation" is. here used in the technical sense to represent the mean of a large number of independent trials, or combinations, in each of which the phases are redistributed at random. It is important to remember that it is infinitely improbable that the expectation will be confirmed in a single trial, however large n may be. Thus in a single combination of many vibrations of arbitrary phase there is about an even chance that the intensity will be less than ^n.
* Phil. Mag. Vol. x. p. 73 (1880); Scientific Papers, Vol. i. p. 491. For another method see Theory of Sound, 2nd ed.  42 a, and for a more complete theory K. Pearson's Math. Contributions to the Theory of Evolution, xv, Dulau, London. 0) = 0, viz. %z cos 0=0, 5 '135, 8-417, 11-620, etc.    W. P. S.]